Lebesgue-type inequalities in sparse sampling recovery

TL;DR

This paper extends Lebesgue-type inequalities for sparse sampling recovery from the square norm to general $L_p$ norms, using a greedy algorithm in Banach spaces, and explores stable recovery linked to sampling discretization.

Contribution

It introduces a Weak Chebyshev Greedy Algorithm for $L_p$ norms in Banach spaces, broadening the scope of sparse recovery techniques beyond Hilbert spaces.

Findings

Effective recovery in $L_p$ norm for $2 \\le p<\\infty$ using the proposed algorithm.

Establishment of connections between stable recovery and sampling discretization.

Extension of Lebesgue-type inequalities to Banach spaces for various integral norms.

Abstract

Recently, it has been discovered that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error being measured in the square norm. It was established that a simple greedy type algorithm -- Weak Orthogonal Matching Pursuit -- based on good points for universal discretization provides effective recovery in the square norm. In this paper we extend those results by replacing the square norm with other integral norms. In this case we need to conduct our analysis in a Banach space rather than in a Hilbert space, making the techniques more involved. In particular, we establish that a greedy type algorithm -- Weak Chebyshev Greedy Algorithm -- based on good points for the $L_p$-universal discretization provides good recovery in the $L_p$ norm for $2\le p<\infty$. Furthermore, we discuss the problem of stable recovery and demonstrate its close relationship with sampling discretization.